Section 10.3 – The Inverse of a Matrix

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The 2 x 2 Inverse

dA

a b

c

Find the inverse of

d

adet A

b

c

1

OR

1

ad bc c

d

a

b

1. Find det A

a. If det A = 0, no inverse

b. If , continue to 2.det A 0

2. Switch elements of main diagonal (blue elements)

3. Change signs of off diagonal (red elements)

d

adet A

b

c

1

Find the inverse of 2 3

1 5

1

10 3 1

5

2

3

3

1

5

2

1

7

Find the inverse of 4 3

5 6

1

24 15 5

6

4

3

3

5

6

4

1

9

Find the inverse of each of the following matrices:

7 2

3 2

0 3

5 4

6 3

4 2

2 21

20 3 7

4 31

15 5 0

NO INVERSE12 – 12 = 0

The 3 x 3 Inverse

1. Find the determinant of the matrix (given)

a. If det A = 0, no inverse.

b. If , continue to 2.det A 0

2. Find the values of the NINE minors.

3. Place +/- into the resulting matrix. (diamond)

4. “Transpose” the matrix (switch rows and columns)

5. Write final answer.

Find the inverse of A if

3 1 7

A 2 0 3 and det A 12

1 1 4

0 3 2 3 2 0

1 4 1 4 1 1

1 7 3 7 3 1

1 4 1 4 1 1

1 7 3 7 3 1

0 3 2 3 2 0

Step 2

Step 1

3 11 2

3 5 2

3 23 2

3 11 2

3 5 2

3 23 2

Step 2 Answer:

Step 3

3 2

5

3

11

3 2

23 2

3 11 2

3 5 2

3 23 2

Step 4

T3 11 2

3 5 2

3 23 2

3 3 3

11 5 23

2 2 2

Step 53 3 3

111 5 23

122 2 2

Find the inverse of A if

4 1 2

A 0 3 1 and det A 65

1 0 6

Complete Step 2 only. You need not show work.

3 1 0 1 0 3

0 6 1 6 1 0

1 2 4 2 4 1

0 6 1 6 1 0

1 2 4 2 4 1

3 1 0 1 0 3

18 1 3

6 22 1

7 4 12

Step 2 Answer

18 1 3

6 22 1

7 4 12

Complete Steps 3, 4 and 5.

Step 3 Step 4 Step 5

1

6 1

4

18 3

22

7 12

18 6 7

1 22 4

3 1 12

18 6 71

1 22 465

3 1 12

Find the inverse of A if

2 0 3

A 4 6 1 and det A 46

2 1 0

Complete Step 2 only. You need not show work.

6 1 4 1 4 6

1 0 2 0 2 1

0 3 2 3 2 0

1 0 2 0 2 1

0 3 2 3 2 0

6 1 4 1 4 6

1 2 16

3 6 2

18 10 12

Step 2 Answer

Complete Steps 3, 4 and 5.

1 2 16

3 6 2

18 10 12

Step 3 Step 4 Step 5

2

3 2

10

1 16

6

18 12

1 3 18

2 6 10

16 2 12

1 3 181

2 6 1046

16 2 12

Find the inverse of each of the following matrices:

4 1

3 6

2 1

5 7

0 1 3

2 4 1 if det A 34

6 5 1

6 11

21 3 4

7 11

9 5 2

9 8 14

16 18 6

11 6 2

9 16 111

8 18 634

14 6 2